chapter 3: functions and graphs 3.2: graphs of functions

Chapter 3: Functions and Graphs3.2: Graphs of FunctionsEssential Question: What can you look for in a graph to determine if the graph represents a function?

3.2: Graphs of Functions• Ex 1: Functions Defined by Graphs

▫A graph may be used to define a function or relation. Suppose that the graph below defines a function f.

▫Find: f (0) f (3) f (2) The domain of f

The range of f

5 10–5–10 x

5

10

–5

–10

y

f (0) = 7

f (3) = 0

f (2) = undefined

[-8, 2) and (2, 7]

[-9, 8]

3.2: Graphs of Functions• Ex 2: The Vertical Line Test

▫ A graph in a coordinate plane represents a function if and only if no vertical line intersects the graph more than once.

▫ Not a Function Function

|4 8–4–8 x

4

8

–4

–8

y

4 8–4–8 x

4

8

–4

–8

y

3.2: Graphs of Functions• Ex 3: Where a Function is

Increasing/Decreasing▫A function is said to be increasing on an

interval if its graph always rises as you move left to right.

▫It is decreasing if its graph always falls as you move left to right

▫A function is said to be constant on an interval if its graph is a horizontal line over the interval

3.2: Graphs of Functions• Ex 3: Where a Function is

Increasing/Decreasing▫On what interval is the function f (x) = |x| + |x

– 2| increasing? Decreasing? Constant? Graph the function It suggests that f is

(For now) Only usebrackets if an intervalis constant

x

y

Decreasing from (-∞, 0)

Increasing on (2, ∞)Constant on [0, 2]

3.2: Graphs of Functions

•Assignment▫Page 160

1 – 14, 17 & 18 (all problems)

Chapter 3: Functions and Graphs3.2: Graphs of FunctionsDay 2Essential Question: What can you look for in a graph to determine if the graph represents a function?

3.2: Graphs of Functions• Ex 4: Finding Local Maxima and Minima

▫A graph of a function may include some peaks and valleys.

▫The peak may not be the highest point, but it is the highest point in its area (called a local maximum)

▫A valley may not be the lowest point, but it is the lowest point in its area (called a local minimum)

▫Calculus is usually needed to find exact local maxima and minima. However, they can be approximated with a calculator.

3.2: Graphs of Functions• Ex 4: Finding Local Maxima and Minima

▫Graph f (x) = x3 – 3.8x2 + x + 1 and find all local maxima and minima. Graph is shown on calculator You can find local maxima and minima by using

the FMIN and FMAX just like finding the root from a graph. [Graph] → [more] → [math] → [fmin]/[fmax]

3.2: Graphs of Functions•Ex 5: Analyzing a Graph

▫Concavity and Inflection Points A point where the curve changes concavity is called

an inflection point An inflection point will be always be between a local

maximum and local minimum’s x-values Concavity is used to describe the way a curve bends

Connect two points on a curve, between inflection points

If the line is above the curve, it’s concave up If the line is below the curve, it’s concave down

▫Open up = concave up, open down = concave down


•Ex 5: Analyzing a Graph▫Graph the function f (x) = -2x3 + 6x2 – x + 3▫Find

All local maxima and minima of the function Intervals where the function is

increasing/decreasing All inflection points of the function Intervals where the function is concave up and

where it is concave down



19-27, 33-39 (odd problems) Hint #1: Do problems 23 – 27 before 19 & 21 Hint #2: For 33 – 35, find the inflection point

first Hint #3: For 37 & 39:

▫I don’t need to see your graph (part “a”)▫Find part “c” before part “b”▫Find part “e” before part “d”

Chapter 3: Functions and Graphs3.2: Graphs of FunctionsDay 3Essential Question: What can you look for in a graph to determine if the graph represents a function?

3.2 Graphs of Functions•Ex 6: Graphing a Piecewise Function

▫To graph a piecewise function by hand Sketch (lightly) each of the graphs Use the individual domain rule to only use the

specified part of the graph & put them together▫To graph a piecewise function on the

calculator Enter the function in normally Divide it by the domain of its piece

Inequality symbols are in the test menu (2nd, 2) Compound inequalities must be split up

3.2: Graphs of Functions•Ex 6: Graphing a Piecewise Function

(calculator)

▫Graph

▫On the graphing calculator (TI-86): x2/(x<1) x+2/((1<x)(x<4))

▫On iPhone apps (Desmos) {x<1 : x2, 1<x<4: x+2}

2 if 1( )

2 if 1 4

x xf x

x x


•Ex 7: The Absolute-Value Function▫Graph f (x)=|x|▫This is also a piecewise function

▫

For the first equation, flip the sign on all terms that were inside the absolute value signs.

Domain is split where the stuff inside the absolute value would equal 0 (the x-coordinate of the vertex of the absolute value function)

if 0( )

if 0

x xf x

x x


•Ex 7: The Absolute-Value Function #2▫Graph f (x)=|2x – 6| + 4

What are the two equations?

Where do the equations split? (Where’s the vertex?)

( )f x

2x – 6 + 4 = 2x - 2

-2x + 6 + 4 = -2x +10

2x – 6 = 0 +6 +6 2x = 6 x = 3

, x > 3

, x < 3


•Ex 8: The Greatest Integer Function▫Graph f (x)=[x]

We enter the function in as “int x” Doesn’t look quite right, does it?

To change graphing type (Only necessary for the greatest integer

function) On the screen to enter functions, press more Press F3 for “Style”, use the (dot display)

setting



41 - 53 (odd problems)

Chapter 3: Functions and Graphs3.2: Graphs of FunctionsUncovered This YearEssential Question: What can you look for in a graph to determine if the graph represents a function?


•Ex 9: Parametric Graphing▫In parametric graphing, both the x and y

coordinate are given functions to a 3rd variable, t.

▫Graph the curve given by x=2t + 1 y = t2 – 3

▫Solution, make a table of values, and sketch


•Ex 9: Parametric Graphing▫x=2t + 1▫y = t2 – 3

▫Now graph

t x = 2t + 1

y = t2 - 3

(x, y)

-2 -3 1 (-3, 1)

-1 -1 -2 (-1, -2)

0 1 -3 (1, -3)

1 3 -2 (3, -2)

2 5 1 (5, 1)

3 7 6 (7, 6)

3.2: Graphs of Functions•Ex 10: Graphing (w/ calc) in parametric mode

▫Change mode (2nd, mode) to “Param” (5th down)▫Now when you go to graph, y(x) is changed to E(t)

You also now enter in two functions at a time (x & y) To graph y = f (x) in parametric mode

Let x = t and y = f (t) To graph x = f(y) in parametric mode

Let y = t and x = f (t) Alter your window

Change the t-step = 0.1

3.2: Graphs of Function

•Ex 10: Graphing in Parametric Mode

1)Graph

▫Let x = t and

2)Graph x = y2 – 3y + 1

▫Let y = t and x = t2 – 3t + 1

21

32

xy

21

32

ty

chapter 3: functions and graphs 3.2: graphs of functions

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